A176411 A symmetrical triangle of leading ones adjusted polynomial coefficients based on Hermite orthogonal polynomials: t(n,m)=CoefficientList[HermiteH[n, x], x][[m + 1]] + Reverse[CoefficientList[ HermiteH[n, x], x]][[m + 1]] - (CoefficientList[HermiteH[n, x], x][[1]] + Reverse[CoefficientList[HermiteH[n, x], x]][[1]]) + 1.
1, 1, 1, 1, -1, 1, 1, -19, -19, 1, 1, -27, -123, -27, 1, 1, 89, -191, -191, 89, 1, 1, 57, 297, 57, 297, 57, 1, 1, -1807, -1471, 3233, 3233, -1471, -1807, 1, 1, -1935, -18959, -1935, 24945, -1935, -18959, -1935, 1, 1, 29729, -9727, -81151, 47873, 47873, -81151
Offset: 0
Examples
{1},
{1, 1},
{1, -1, 1},
{1, -19, -19, 1},
{1, -27, -123, -27, 1},
{1, 89, -191, -191, 89, 1},
{1, 57, 297, 57, 297, 57, 1},
{1, -1807, -1471, 3233, 3233, -1471, -1807, 1},
{1, -1935, -18959, -1935, 24945, -1935, -18959, -1935, 1},
{1, 29729, -9727, -81151, 47873, 47873, -81151, -9727, 29729, 1},
{1, 29217, 308577, 29217, -212703, 29217, -212703, 29217, 308577, 29217, 1}
Programs
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Mathematica
t[n_, m_] := CoefficientList[HermiteH[n, x], x][[m + 1]] + Reverse[CoefficientList[ HermiteH[n, x], x]][[m + 1]] - (CoefficientList[HermiteH[n, x], x][[1]] + Reverse[CoefficientList[HermiteH[n, x], x]][[1]]) + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
Formula
t(n,m)=CoefficientList[HermiteH[n, x], x][[m + 1]] + Reverse[CoefficientList[ HermiteH[n, x], x]][[m + 1]] - (CoefficientList[HermiteH[n, x], x][[1]] + Reverse[CoefficientList[HermiteH[n, x], x]][[1]]) + 1
Comments